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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Attouch-Wets convergence and a differential operator for convex functions


Authors: Gerald Beer and Michel Théra
Journal: Proc. Amer. Math. Soc. 122 (1994), 851-858
MSC: Primary 49J45; Secondary 46N10, 49J50
MathSciNet review: 1204368
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Abstract: The purpose of this note is to characterize Attouch-Wets convergence for sequences of proper lower semicontinuous convex functions defined on a Banach space X in terms of the behavior of an operator $ \Delta $ defined on the space of such functions with values in $ X \times R \times {X^ \ast }$, defined by $ \Delta (f) = \{ (x,f(x),y):(x,y) \in \partial f\} $. We show that $ \langle {f_n}\rangle $ is Attouch-Wets convergent to f if and only if points of $ \Delta (f)$ lying in a fixed bounded set can be uniformly approximated by points of $ \Delta ({f_n})$ for large n. The operator $ \Delta $ is a natural carrier of the Borwein variational principle, which is a key tool in both directions of our proof.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1204368-5
PII: S 0002-9939(1994)1204368-5
Keywords: Convex function, Attouch-Wets convergence, subdifferential, Borwein variational principle, regularization
Article copyright: © Copyright 1994 American Mathematical Society